Thanks for the heads up, sorry for the late response (WP does NOT email developers when someone posts a question here).
I don't say this plugin is compatible with WPMU (it's a lot of work for me to setup and maintain a WPMU sandbox just to test plugins I'm giving away, so I stopped doing it). However, AUTH_COOKIE is defined at /wp-settings.php and your error is being generated in wp-includes/pluggable, so it seems unlikely it has anything specifically to do with my plugin.
It almost sounds like files are not loading in the correct order, so AUTH_COOKIE isn't being defined before something else that needs it calls it. It could also be something to do with a combination of plugins (unless you tell me this was the ONLY active plugin at the time), or some other issue I can not even guess... Just for fun, you could try adding a line of code at the top of my main plugin file that says
if ( !defined('AUTH_COOKIE') ) require_once('/www/vhosts/wordpress-stable-mu.local/wp-settings.php'); and see if that solves the issue for you (if it does, please let me know).
If/when I have paying clients that are using WPMU (and therefore other reasons to set up and maintain a WPMU sandbox environment) I may revisit this issue, but for now I'm going to mark it resolved, since the plugin isn't currently tested for use in a WPMU environment and so far (fingers crossed) you are the only person who has reported this issue.